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25
Online Conflictfree Colorings for Hypergraphs
, 2007
"... We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically ..."
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We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CFcoloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results. We introduce deterministic online CFcoloring algorithms for points on the line with respect to intervals and for points on the plane with respect to halfplanes (or unit discs) that use Θ(log n) colors and recolor O(n) points in total.
ConflictFree Coloring and its Applications
, 2010
"... Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to c ..."
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Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols and several other fields, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.
ConflictFree Coloring of Points on a Line with respect to a Set of Intervals
"... We present a 2approximation algorithm for CFcoloring of points on a line with respect to a given set of intervals. The running time of the algorithm is O(nlog n). 1 ..."
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Cited by 5 (0 self)
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We present a 2approximation algorithm for CFcoloring of points on a line with respect to a given set of intervals. The running time of the algorithm is O(nlog n). 1
Uniquemaximum and conflictfree colorings for hypergraphs and tree graphs, arXiv:1002.4210v1
, 2010
"... We investigate the relationship between two kinds of vertex colorings of hypergraphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every hyperedge of the hypergraph the maximum color appears only once. In a conflictfree coloring, ..."
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Cited by 4 (1 self)
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We investigate the relationship between two kinds of vertex colorings of hypergraphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every hyperedge of the hypergraph the maximum color appears only once. In a conflictfree coloring, in every hyperedge of the hypergraph there is a color that appears only once. We concentrate in hypergraphs that are induced by paths in tree graphs.
Conflictfree colorings of graphs and hypergraphs
"... A coloring of the vertices of a hypergraph H is called conflictfree if each hyperedge E of H contains a vertex of “unique ” color that does not get repeated in E. The smallest number of colors required for such a coloring is called the conflictfree chromatic number of H, and is denoted by χCF(H). ..."
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A coloring of the vertices of a hypergraph H is called conflictfree if each hyperedge E of H contains a vertex of “unique ” color that does not get repeated in E. The smallest number of colors required for such a coloring is called the conflictfree chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even et al. (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyze this notion for general hypergraphs. It is shown that χCF(H) ≤ 1/2 + √ 2m + 1/4, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m 1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász’s Local Lemma, the same result holds for hypergraphs, in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial time algorithms to obtain such colorings. Our machinery can also be applied to the hypergraphs induced by the neighborhoods of the vertices of a graph. It turns out that in this case we need much fewer colors. For example, it is shown that the vertices of any graph G with maximum degree ∆ can be colored with log 2+ǫ ∆ colors, so that the neighborhood of every vertex contains a point of “unique ” color. We give an efficient deterministic algorithm to find such a coloring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need (1) to correct a small error in the MolloyReed approach; (2) to restate and reprove their result in a deterministic form.
Online conflict free coloring for halfplanes, congruent disks, and axisparallel rectangles
, 2008
"... We present randomized algorithms for online conflictfree coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearlyequal axisparallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability. We also present a d ..."
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We present randomized algorithms for online conflictfree coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearlyequal axisparallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability. We also present a deterministic algorithm for online CF coloring of points in the plane with respect to nearlyequal axisparallel rectangles, using O(log³ n) colors. This is the first efficient (that is, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.
ON THE CHROMATIC NUMBER OF GEOMETRIC HYPERGRAPHS
 VOL. 21, NO. 3, PP. 676–687
, 2007
"... A finite family R of simple Jordan regions in the plane defines a hypergraph H = H(R) where the vertex set of H is R and the hyperedges are all subsets S ⊂Rfor which there is a point p such that S = {r ∈Rp ∈ r}. The chromatic number of H(R) is the minimum number of colors needed to color the membe ..."
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Cited by 3 (0 self)
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A finite family R of simple Jordan regions in the plane defines a hypergraph H = H(R) where the vertex set of H is R and the hyperedges are all subsets S ⊂Rfor which there is a point p such that S = {r ∈Rp ∈ r}. The chromatic number of H(R) is the minimum number of colors needed to color the members of R such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs and obtain the following results: (i) Any hypergraph that is induced by a family of n simple Jordan regions such that the maximum union complexity of any k of them (for 1 ≤ k ≤ m) is bounded by U(m) and U(m) m is a nondecreasing function is O ( U(n))colorable. Thus, for example, we prove that any finite family of pseudodiscs can n be colored with a constant number of colors. (ii) Any hypergraph induced by a finite family of planar discs is four colorable. This bound is tight. In fact, we prove that this statement is equivalent to the fourcolor theorem. (iii) Any hypergraph induced by n axisparallel rectangles is O(log n)colorable. This bound is asymptotically tight. Our proofs are constructive. Namely, we provide deterministic polynomialtime algorithms for coloring such hypergraphs with only “few ” colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs). As an application of (i) and (ii) we obtain simple constructive proofs for the following: (iv) Any set of n Jordan regions with near linear union complexity admits a conflictfree (CF) coloring with polylogarithmic number of colors. (v) Any set of n axisparallel rectangles admits a CFcoloring with O(log2 (n)) colors.
Graph uniquemaximum and conflictfree colorings
 In Proc. 7th International Conference on Algorithms and Complexity (CIAC
, 2010
"... We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflictfree coloring, in every path ..."
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We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflictfree coloring, in every path of the graph there is a color that appears only once. We also study computational complexity aspects of conflictfree colorings and prove a completeness result. Finally, we improve lower bounds for those chromatic numbers of the grid graph.